Mathematical Problems in Engineering

Volume 2015, Article ID 360783, 8 pages

http://dx.doi.org/10.1155/2015/360783

## Design of a Discrete Tracking Controller for a Magnetic Levitation System: A Nonlinear Rational Model Approach

^{1}School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China^{2}The Bristol Institute of Technology, University of the West of England, Bristol BS 161QY, UK

Received 5 June 2014; Accepted 27 August 2014

Academic Editor: Kang Li

Copyright © 2015 Fernando Gómez-Salas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work proposes a discrete-time nonlinear rational approximate model for the unstable magnetic levitation system. Based on this model and as an application of the input-output linearization technique, a discrete-time tracking control design will be derived using the corresponding classical state space representation of the model. A simulation example illustrates the efficiency of the proposed methodology.

#### 1. Introduction

The magnetic levitation system is an interesting nonlinear and unstable complex system. Due to its great importance in many fields of the engineering, this system is becoming popular in recent years. In fact, the magnetic levitation system has been successfully applied in high speed trains [1, 2], frictionless bearings [3, 4], and vibration isolation tables [5].

Although the magnetic levitation has been successfully applied to many real systems which work in continuous time, most of the control functions need to be implemented through digital devices such as computers. For this reason, a direct design strategy is to design discrete-time controllers directly from discrete-time models based on either input-output models or state space models. Moreover, when a nonlinear plant needs to be controlled, this immediately raises the problem of what class of model should be used [6]. Polynomial models are generally used for many applications but they are inadequate for severe nonlinear systems and the nonlinear rational NARMAX model was introduced to overcome this problem. The main advantage of the rational model is the efficiency to depict high nonlinearities with a few parameters. However, control design and identification for this model are comparatively complicated compared with the polynomial models [7–10].

Despite the fact that many works have used either input-output models or state space models for control design, the model based control system design expresses a clear preference for the latter. In fact, the classical state space representation is still dominant in the control literature since it allows describing internal dynamics in almost all systems (mechanical systems, electrical systems, economics systems, and so forth). Some advantages of this representation can be described as follows: the model directly provides a time-domain solution, which is ultimately the thing of interest; the form of the solution is the same as that for a single first-order differential equation; the effect of initial conditions can be easily incorporated in the solution; the matrix-vector modeling is very efficient in computation and computer implementation, which are particularly significant in large and complex system simulations. In this way, the importance of the state space models for either simulation or control design is undisputed.

As is well known, the nonlinear trajectory tracking problem is a topic of great importance in many real applications [11–13]. In practical control problems, however, the nonlinearities of a complex system are in general very difficult to handle in a direct way. Therefore, a well-known control strategy consists, first of all, in transforming the system structure by appropriate feedbacks, so as to substitute nonlinear relations with linear ones [14].

In the literature, a number of control strategies have been reported in order to design stabilizing control laws for the magnetic levitation system. However, these existing solutions are mainly discussed using classical methods [15, 16]. Moreover, new approaches have been reported (frequently based on continuous-time techniques); however, these are based on the linearized model about a nominal operating point and the tracking performance falls quickly when the deviations leave the nominal operating point [17]. Therefore, to ensure long ranges of motion and still having a good tracking performance, it is necessary to consider a nonlinear model rather that a linear one.

In this spirit and since control functions need to be implemented through digital devices, this work considers the problem of finding a discrete-time nonlinear rational model for a generic magnetic levitation system and then designing a discrete-time tracking controller for this model.

The arrangement of this study is as follows. In Section 2, the discrete-time model for a generic magnetic levitation system is presented. Section 3 formulates the tracking problem studied in this work. Additionally, this section presents the corresponding feedback solution derived as an application of the input-output linearization technique. In Section 4, a numerical example is presented in order to validate the proposed methodology of the study. Finally, in Section 5 some conclusions are given.

#### 2. System Dynamics and Modeling

Figure 1 shows the schematic of a generic magnetic levitation system. The target of this system is to control the position of the small ball of mass . By using an electromagnetic force , which is produced from a current , this small metal ball can be displaced a distance from the electromagnet. Notice, however, that this system assumes that .